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## 4Random Walks

Let X1, X2, . . . be i.i.d. taking values in Rd and let Sn = X1 + ··· + Xn. Sn is a random walk. In the previous chapter, we were primarily concerned with the distribution of Sn. In this one, we will look at properties of the sequence S1(ω), S2(ω), . . . For example, does the last sequence return to (or near) 0 infinitely often? The first section introduces stopping times, a concept that will be very important in this and the next two chapters. After the first section is completed, the remaining three can be read in any order or skipped without much loss. The second section is not starred since it contains some basic facts about random walks.

### 4.1 Stopping Times

Most of the results in this section are valid for i.i.d. X’s taking ...

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